Say I have set of $m$ Boolean vectors $$B = \{x_1,\ldots, x_m\}$$ where $x_i \in \{0,1\}^n$.

We know the following about the vectors $x_i \in B$:

(i) $\|x_i\| \in [1,n-1]$ for all $x_i \in B$ (*at least 1 non-zero coordinate and one zero coordinate)*

(ii) $d(x_i, x_j) \geq 1$ for all $x_i, x_j \in B$ (*each vector is distinct*)

(iii) $d(x_i, x_j) \leq r$ for all $x_i, x_j \in B$ (*all vectors differ by at most $r$ coordinates*)

Here $d(x_i, x_j) = \|x_i - x_j\| = \sum_{k=1}^{n} 1[x_{i,k} \neq x_{j,k}]$ is the Hamming distance between $x_i$ and $x_j$.

**My question is the following:** Given a vector $x_i \in B$, what is the maximum number of coordinates where $x_i$ is false, but at least $t$ of the other vectors are true? In other words, I am looking for an upper bound on the size of the following set:

$$Z(x_i, t) = \left\{k =1,\ldots, n ~\Big|~ x_{i,k} = 0 ~\text{and}~ \sum_{j \neq i} x_{j,k} \geq t \right\} $$

for values of $t \in [1, |B| - 1]$.

**Note**: I can derive the following upper bound when $t = 1$ (see a related post for a simpler problem on math.SE):

$$|Z(x_i, 1)| \leq \frac{r - \|x_i\| + \max_{j \neq i} \|x_j\|}{2}$$

However, I'm not sure how to extend this to settings where $t \geq 2$ (or whether this is actually the best bound I could produce for $t = 1$). Any help would be appreciated.